Transcription of his NSW birth certificate ex FindMyPast:

First name(s) William
Last name Cartwright
Birth year 1811
Registration year 1811
Denomination Church of England
Registration district Windsor, St Matthew's
State New South Wales
Country Australia
Father's first name(s) Robert
Mother's first name(s) Mary
Volume reference V18112558 1A
Record set New South Wales Births

The actual certificate gives his birth and baptism dates.
________________

There is a record on FindMypast for a marriage in Tasmania between Wm Cartwright and Elizabeth Martin in 1844.  But it does not include any information about either of their parents so it cannot be used until there is greater confidence that the marriage is of this William Cartwright:

First name(s) William
Last name Cartwright
Birth year 1811
Residence Tasmania, Australia
Marriage year 1844
Marriage date 02 Jan 1844
Spouse's first name(s) Elizabeth
Spouse's last name Martin
Marriage place Richmond, Tasmania, Australia
State Tasmania
Country Australia
Groom's age 33
Groom's birth date 1811
Bride's age 22
Bride's birth date 1822
Record set Australia, Marriages, 1810-1980

Nevertheless the fact that his slightly elder brother Thomas did marry and then live in Tasmania, does make this record tempting for William.
__________________